intech-computation of transient near field radiated by electronic devices from frequency data

7/28/2019 InTech-Computation of Transient Near Field Radiated by Electronic Devices From Frequency Data

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Computation of Transient Near-Field Radiatedby Electronic Devices from Frequency Data

Blaise Ravelo and Yang LiuIRSEEM (Research Institute in Embedded Electronic System), EA 4353,

Graduate School of Engineering ESIGELEC,76801 Saint Etienne du Rouvray Cedex,

France

1. Introduction

Facing to the increase of architecture complexity in the modern high-speed electronicequipments, the electromagnetic compatibility (EMC) characterization becomes a crucialstep during the design process. This electromagnetic (EM) characterization can manifestwith the unintentional conducting or radiating perturbations including, in particular, thenear-field (NF) emissions. Accurate modelling method of this emission in NF zone becomesone of electronic engineer designers and researchers most concerns (Shi et al. 1989, Baudryet al. 2007, Vives-Gilabert et al. 2007, Vives-Gilabert et al. 2009, Song et al. 2010, Yang et al.2010). This is why since the middle of 2000s; the NF modelling has been a novel speciality of

the electronic design engineers. This modelling technique enables a considerable insuranceof the reliability and the safety of the new electronic products. To avoid the doubtful issuesrelated to the EM coupling, this analysis seems indispensable for the modern RF/digitalelectronic boards vis--vis the growth of the integration density and the operating numericaldata-speed which achieves nowadays several Gbit/s (Barriere et al. 2009, Archambeault etal. 2010). In this scope, the influence of EM-NF-radiations in time-domain and in ultra-wideband (UWB) RF-/microwave-frequencies remains an open-question for numerous electronicresearchers and engineer designers (Ravelo et al. 2011a & 2011b, Liu Y. et al. 2011a & 2011b).In the complex structures, the current and voltage commutations in the non-linear electronicdevices such as diodes, MOSFETs and also the amplifiers can create critical undesiredtransient perturbations (Jauregui et al. 2010a, Vye 2011, Trscher 2011, Kopp 2011). Such

electrical perturbations are susceptible to generate transient EM-field radiations which needto be modelled and mastered by the electronic handset designers and manufacturers.

1.1 Overview on the NF radiations characterization occurring in the RF/microwave-device in time-domain

It is noteworthy that the frequency-investigations on the EM-radiation of electronic devicesare not sufficient for the representation of certain EM-transient phenomena notably whenthe sources of perturbations behave as a short duration pulse-wave. In fact, it does notenable to precise the probably instant times and the intensity peak of the EM-pulse. That iswhy the time-domain representation is particularly essential for the infrequent and ultra-


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Fourier Transform Applications4

short duration wave emission analysis. In order to investigate more concretely theunwanted time-domain perturbations, different EM-NF modelling and measurementtechniques were recently introduced and published in the literature (Cicchetti 1991, Adada2007, Liu L. et al. 2009, Winter & Herbrig 2009, Ordas et al. 2009, Braun et al. 2009, Rioult et

al. 2009, Xie & Lei 2009, Edwards et al. 2010, Jauregui et al. 2010b, Ravelo 2010).Furthermore, several EM-solvers are also integrated in the commercial simulation tools forthe determination of the EM-field radiations by the RF/microwave devices especially infrequency domain (ANSOFT 2006, AGILENT 2008, ANSYS 2009, NESA 2010).

Currently, the computation method of the EM-field becomes systematically more and morecomplicated when the electronic systems operate with baseband UWB signals. Despite therecent investigations conducted on the finite-difference time-domain method (FDTD)method (Liu et al. 2009, Jauregui et al. 2010b), the accuracy of the computation results withthese time-domain commercial tools remains difficult to evaluate when the perturbationsources are induced from ultra-short duration transient NF. In addition, more practical

techniques (Cicchetti 1991, Braun et al. 2009, Winter & Herbrig 2009, Ordas 2009, Rioult et al.2009) have been also introduced for the measurement of the electric- and electronic- systemelectromagnetic interference (EMI). But compared to the existing frequency measurementtechniques, they are much better because of the limitations either in terms of space-resolution or electro-sensitivity or simply the calibration process. So, the evaluation of theaccurate graphs of time-dependent EM-waves in NF is still an open challenge.

To cope with this limitation, in this chapter, an efficient computation methodology based onthe transformation of wide bandwidth and baseband frequency-dependent data for thedetermination of the transient EM-NF mapping permitting is developed. In order to takeinto account the transient radiations specific to the expected use cases, an adequate

excitation signal should be considered. This excitation is usually defined according tocertain technical parameters (amplitude, temporal width, variation speed, time-duration)which qualifies the undesired disturbing signal susceptible to propagate in the emittingcircuits. Then, the fast Fourier transform (fft) mathematical treatment of the assumeddisturbing signal synchronized with the given discrete frequency-dependent data in theadequate frequency range enables to determine the transient wave radiation mapping.

1.2 Background on the EMC application of the transient EM-NF

As aforementioned and discussed (Rammal et al. 2009, Jauregui et al. 2010a), the EM-transientanalysis is actually important for the immunity predictions in the mixed or analogue-digital

components constituting the high-speed electronic boards regarding the eventual radiationsof high power electrical circuitry as the case of neighbouring hybrid electric vehiclepropulsion systems. To assess such an EMC effect, as reported in (Adada 2007), theelectronic circuit designers working on analogue/mixed signal (AMS) subsystems havepreferred software tools such as SPICE, while those working on RF/microwave front-endcomponents have tended to manipulate S-parameter frequency-domain design andsimulation tools. By cons, currently, the fusion of the both approaches as AMS engineers arerequired to make further analysis on the critical components is needed by using theadequate EM simulation tools. In this case, we have to elaborate the context of ultra-wideband (UWB). Currently, this topic is one of improvement techniques in the area of EMCapplication. In this optic, the modelling of mixed component EM-NF emission becomes one


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Computation of Transient Near-Field Radiated by Electronic Devices from Frequency Data 5

of the crucial steps before the implementation process. Therefore, the undesired EMCradiations should be investigated not only in frequency-domain but also in time-domain.For this reason, we propose in this chapter an extraction method enabling to determine thetime-domain EM-NF maps from the frequency-dependent data by using the Fourier

transform of the 2D data.

1.3 Outline of the presented chapter

To make this chapter better to understand, it is organized in three main sections. Section 2describes the methodology of the time-frequency computation-method proposed. It detailshow to extract the transient EM-NF radiation from the given time-dependent excitationsampled signal and the frequency-dependent data. Then, more concrete validation of thecomputation-method investigated by considering the EM-NF radiated by an arbitrary set ofmagnetic dipoles is devoted in Section 3. The EM-NF reference data are calculated with thetheoretical formulas introduced in (Baum 1971 & 1976, Singaraju & Baum 1976). As

reviewed by certain research works (Hertz 1892, Chew & Kong 1981, Lakhtakiaa et al. 1987,Song & Chen 1993, Jun-Hong et al 1997, Schantz 2001, Selin 2001, Smagin & Mazalov 2005,Sten & Hujanen 2006, Ravelo 2010), the analytical calculation performed with the EM-waveemitted by elementary dipoles allows to realize more practical and more explicitmathematical analyses of the EM-field expressions in different physic areas. We point outthat the EM-field emitted by electronic devices can be modelled by the radiations of theoptimized combination of elementary EM-dipoles (Fernndez-Lpez et al. 2009). To confirmthe feasibility of the method proposed, an application with another proof of concept with aconcrete electronic device is also offered in Section 4. This practical verification will be madetoward a microwave electronic design of low-pass planar microstrip filter operating up untilsome GHz. Lastly; Section 5 draws the conclusion of this chapter.

2. Methodology of the time-frequency computation method investigated

The present section is divided in two different parts. First, an explicit description illustrateshow to examine the transient excitation signal for the UWB applications. Afterward, thedevelopment of the routine process indicating the algorithm of the computation methodproposed is elaborated.

2.1 Frequency coefficient extraction

Let us denote i(t) the transient current which is considered also as the excitation of the under

test electronic structure. The sampled data corresponding to this test signal is supposeddiscretized from the starting time tmin to the stop time tmax with time step equal to t. In thiscase, the number n of time-dependent samples is logically, equal to:

max minintt t

nt

=

, (1)

with int(x) expresses the lowest integer number greater than the real x. Accordingly, via thefast Fourier transform (fft), the equivalent frequency-dependent spectrum of i(tk) (with tk =k.t and k = {1n}) can be determined. The frequency data emanated by this mathematicaltransform are generally as a complex number denoted by [ ]( ) ( )k kI f fft i t= . Therefore, the


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magnitude of this signal spectrum from DC to a certain frequency in function of the initialtime-step sampled data can be extracted as depicted in Fig. 2. Consequently, by denoting I0the sinusoidal current magnitude for generating the magnetic field spectrum 0( )H f , thefollowing complex coefficients of the input current in function of frequency as illustrated in

Fig. 1:

0

( )stepk

I k fc

I

= . (2)

Fig. 1. Extraction of the frequency coefficients from the excitation signal spectrum

It corresponds to the discrete data at each sample of frequencyfk = k.fstep for k = {1n} havinga step-frequency given by:

max min

1stepf

t t=

. (3)

We underline that this method requires a frequency range [fmin, fmax] whose the lowestfrequency value fmin of ( )I f -data is equal to the step frequency fstep. It means that thespectrum value can be extrapolated linearly to generate the excitation signal steady-statecomponent atf= 0. In practice, it does not change the calculation results because accordingto the signal processing theory, the DC-component of transient waves with ultra-short time

duration at very low frequencies is usually negligible. The upper frequency fmax shouldcorrespond to the frequency bandwidth containing 95-% of the excitation signal consideredspectrum energy.

2.2 Routine process of the computation-method proposed

The computation-method under study is mainly consisted of two different steps. The firststep is focused on the time-domain characterization of the excitation current considered i(tk)in the specific interval range varying from tmin to tmax with step t. In this first step, thecomplex frequency-coefficients kc should be extracted through the fft-operation asexplained in previous subsection. The following second step is the conversion of the


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frequency-dependent magnetic- or H-field expressed by 00( , , , )H x y z f which is recorded atthe point 0( , , )M x y z chosen arbitrarily, into a time-dependent data denoted by 0( , , , )H x y z t by using the ifft-operation. These points 0( , , )M x y z belong in the X-Y plane positioned atz = z0. In this case, the frequency range considered varying fromfmin tofmax and the frequency

step fstep of 0 0 0( ) ( , , , )H f H x y z f = must be well-synchronized with that of the excitationsignal frequency coefficients ck. Under this condition, the time-dependent data desired

0( ) ( , , , )H t H x y z t= which is generated by the specific excitation signal i(t) can be calculatedwith the inverse fast Fourier transform (ifft) of the convolution product between

0( ) ( )/c f I f I = and 0( )H f written as:

{ }0( ) [ ( ). ( )]H t e ifft c f H f = . (4)

with { }e x represents the real part of the complex number x. Fig. 2 depicts the flow workhighlighting the different operations to be fulfilled for the achievement of the transient EM-field computation-method proposed. This method enables to provide the time-dependent

Fig. 2. Flow work illustrating the transient H-field radiation computation-method proposedknowing the temporal range, tmin and tmax stept of the excitation signal i(tk) and also thefrequency-dependent H-field 0( )H f (here the under bar indicates the complex variables)


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H-field here denoted as H(t) according to the arbitrary form of the excitation and alsoknowing the frequency-dependent H-field data in the frequency range starting from thelowest value to the upper frequency limit equal to the inverse of the time-step t of thediscrete data x(tk).

3. Validation with the radiation of a set of magnetic dipoles

This section presents the verification of the computation method established regarding theradiation of magnetic dipoles shown in Fig. 3 in time-domain. The results were realized byconsidering the frequency-dependent EM-NF-field directly calculated from predefinedanalytical formulas (Baum 1971 & 1976, Singaraju & Baum 1976, Balanis 2005). After theanalytical description of the considered dipole source and also the mathematical definitionsof the H-field expressions in frequency- and time-domains, we will explore the numericalcomputation with Matlab programming.

Fig. 3. Configuration of the elementary magnetic dipole formed by a circular loop withradius a placed at the centre O(0,0,0) of the cartesian (x,y,z)-coordinate system

In this figure, the magnetic dipole assumed as a circular wire having radius a is positionedat the origin of the system. Then the considered point M where the EM-field will beevaluated can be referred either in cartesian coordinate (x,y,z) or in spherical coordinate(r,,). By assuming that the magnetic loop depicted in Fig. 3 is fed by transient currentdenoted i(t), we have the H-field expressions recalled in appendix A (Baum 1971 & 1976,

Singaraju & Baum 1976).

3.1 Description of the radiation source considered

Fig. 4 displays the arbitrary placement representing the set of eight magnetic dipoles in thehorizontal plane Oxy considered for the validation of the calculation method under study.To generate the various behaviours of EM-field graphs, the dipole axes are randomlyoriented as follows:M1,M4,M5 andM8 along Ox-axis,M2 andM6 along Oy, andM3 andM7along Oz. These dipoles are in this case, flowed by an ultra-short duration transient currenti(t) which is considered as a pulse signal. These elementary dipoles are supposed formed bywire circular loops having radius a = 0.5 mm.


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Note that in the present study, the elementary dipoles are supposed ideal, thus, there is nocoupling between each other. By reason of linearity, the total EM-field at any pointM(x,y,z)is consequently the sum of each dipole contribution:

8

1

( , , ) ( , , )kM

k

H x y z H x y z=

= JJG JJG

. (5)

After implementation of the computation algorithm introduced in Fig. 2 in Matlab program,

we obtain the EM-calculation results presented in the next subsection.

Fig. 4. Configuration of the radiating source considered which is comprised of magneticdipoles placed in the horizontal plane xy

3.2 Validation results

In this subsection, comparison results between the transient EM-field maps radiated by theelementary dipoles displayed in Fig. 4 from the direct calculation and from the methodproposed are presented.

3.2.1 Description of the excitation signal

In order to highlight the influence of the form and the transient variation of the disturbingcurrents in the electronic structure, the considered short-duration pulse excitation currenti(t) is assumed as a bi-exponential signal analytically defined in appendix B. One points outthat in order to take into account the truncation effect between tmin and tmax, the consideredsampling data from i(t) should be multiplied by a specific time gate. So that accordingly,

each component ( )I should be assumed as a sine cardinal. But here, this effect can benegligible if the assumed time step is well-accurate. The numerical application was made bytaking the current amplitude IM = 1A and the time-constants 1 = 2/2 = 2 ns. So, from theanalytical relation expressed in (B-4), we have 95% 3.07 Grad.s-1. Fig. 5 displays thetransient plot of this current excitation.

The time interval range of signal test was defined from tmin = 0 ns to tmax = 20 ns with stept = 0.2 ns. One can see that this baseband signal presents a frequency bandwidthfmax ofabout 2 GHz, where belongs more than 95-% of the spectrum signal energy. The datacalculated ( ) [ ( )]I fft i t = generates the frequency-coefficient values of i(t) according to therelation expressed in (2) as described earlier in subsection 2.2.


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This height was arbitrarily chosen in order to generate a significant NF effect in the

considered frequency range. The dimensions of the mapping plane were set at Lx = 110 mmand Ly = 100 mm with space-resolution equal to x = y = 2 mm. First, by using the

harmonic expressions of the magnetic field components, the maps of the frequency-

dependent H-field are obtained fromfmin = 0.05 GHz tofmax = 2.50 GHz with step f= 0.05GHz. Fig. 7 represents the corresponding mappings of the H-field component magnitudes at

f0 = 2 GHz.

Fig. 7. Maps of H-field components magnitude obtained at the frequencyf0 = 2 GHz: (a) Hx,

(b) Hy and (c) Hz directly calculated from expressions (A-8), (A-9) and (A-10)

After the Matlab program implementation of the algorithm indicated by the flow chartschematized in Fig. 2, the results shown in Figs. 8(a)-(d) are obtained via the combination of

the frequency-dependent data of the H-field components associated to the frequency-

coefficients of the excitation signal plotted in Fig. 5. One can see that the EM-mapspresenting the same behaviors as those obtained via the direct calculations displayed in Fig.

6 were established. In addition, we compare also as illustrated in Fig. 9 the modulus of the

H-fields from the method under study and from the 3D EM Field Simulator - CST(Computer Simulation Technology). Furthermore, as evidenced by Figs. 10(a)-(c), very good

correlation between the profiles of the H-field components detected in the vertical cut-plane

along Oy and localized at x = 23 mm was observed. To get further insight about the time-dependent representation of the H-field components, curves showing the variations of Hx(t),

Hy(t) and Hz(t) at the arbitrary point chosen of the mapping plane having coordinates (x = 19

mm, y = 35 mm) are plotted in Figs. 11(a)-(c).


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As results, once again, we can find that the H-field components from the frequency data fit

very well the direct calculated ones. As aforementioned, due to the truncation effects, theHx-component presents a slight divergence at the ending time of the signal. This is

particularly due to the numerical noises at the very low value of the EM field as the case of

the x-component which is absolutely twenty times less than the two other components.

In order to prove in more realistic way the relevance of the investigated method, one

proposes to treat the radiation of concrete electronic devices in the next section.

Fig. 8. Maps of H-field components calculated from the time-frequency computation method

proposed for z0 = 6.5 mm: (a) Hx, (b) Hy, (c) Hz

Fig. 9. Comparison of H-field maps modulus |H|(t0 = 2 ns) obtained from the direct

formulae (in left) and from the time-frequency computation method proposed for

z0 = 6.5 mm


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Fig. 10. Comparisons of the H-field component profiles obtained from the time-frequencycomputation method proposed and the direct calculation, detected in the vertical cut-planex = 23 mm: (a) Hx, (b) Hy and (c) Hz


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Fig. 11. Comparisons of the H-field components temporal variation obtained from the time-frequency computation method proposed and the direct calculation, detected in the arbitrarypoint (x = 19 mm, y = 35 mm): (a) Hx, (b) Hy and (c) Hz


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4. Application with the Transient NF emitted by a microstrip device proof-of-concept

To get further insight about the feasibility of the computation method under study, let us

examine the transient EM-wave emitted by an example of more realistic microwave device.This latter was designed with the standard 3-D EM-tools HFSS for generating the frequency-dependent data which is used for the determination of transient H-NF. Then, the simulationwith CST microwave studio (MWS) was performed for the computation of the referenceH-NF mappings in time-domain. As realistic and concrete demonstrator, a low-passTchebychev filter implemented in planar microstrip technology was designed. Its layout topview including the geometrical dimensions is represented in Fig. 12(a).

This device was printed on the FR4-epoxy substrate having relativity permittivity r= 4.4,thickness h = 1.6 mm and etched Cu-metal thickness t = 35 m. The cut cross section of thismicrowave circuit is pictured in Fig. 12(b).

(a)

(b)

Fig. 12. (a) Top view of the CST design of the under test low-pass microstrip filter (b) Cross-section cut of the under test microstrip filter with metallization thickness t = 35 m anddielectric substrate height h = 1.6 mm

After simulations, we realize the results obtained and discussed in next subsections.


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4.1 CST-computation results

The low-pass filter under test was simulated with CST MWS in the time interval range

delimited by tmin = 0 ns and tmax = 20 ns with step t = 0.2 ns. Therefore, the magnetic-field

maps are presented in Fig. 13. These curves are recorded at the arbitrary instant time t0 = 2ns. Note that this structure was excited by the input transient current plotted in Fig. 5. It

results the graphs of H-field components displayed in Fig. 13.

Fig. 13. Maps of transient H-field components detected at t = 2 ns: (a) Hx, (b) Hy and (c) Hzcomputed from the commercial tool CST


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Similar to the previous section, these H-field components were mapped in the horizontal

plane placed at the height z0 = 6 mm above the bottom surface of the considered filter anddelimited by -38 mm < x < 38 mm and -24 mm < y < 24 mm with space-step x = y = 2 mm.

By using the frequency-dependent data convoluted with the excitation test signal, we will

present next that we can regenerate these transient H-field maps.

4.2 Analysis of the results obtained from the transient field computation methodproposed

Fig. 14. Maps of frequency-dependent H-field components magnitude: (a) Hx, (b) Hy and (c)

Hz computed from HFSS atf= 1 GHz


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After HFSS-simulations carried out in the frequency range starting from fmin = 0.05 GHz tofmax = 2.5 GHz with frequency-step f = 0.05 GHz, the maps of H-field componentmagnitudes displayed in Fig. 14 are recorded. These field components are mapped in thesame horizontal plane as in the previous subsection by taking the height z0 = 6 mm. Among

the series of the field maps obtained, we show here the data simulated at the frequencyf= 1GHz. According to the flow work depicted in Fig. 2, the frequency-dependent data Hx(f),Hy(f) and Hz(f) are employed for the determination of the time-dependent data Hx(t), Hy(t)and Hz(t) regarding the transient input current plotted earlier in Fig. 5. So that byapplication of the computation algorithm under investigation, the H-field maps calculatedfrom Matlab are respectively, depicted in Figs. 15 at the instant time t = 2 ns.

Fig. 15. Maps of H-field components obtained from the proposed method and regarding thesimulated frequency-dependent data from HFSS: (a) Hx, (b) Hy and (c) Hz


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Despite the slight difference of Hx-maps, we observe that the maps are perfectly well-

correlated to those introduced in Fig. 13 of subsection 4.1. For the further smart illustrationof the results correlation, comparisons of H-field profiles calculated with the method

proposed (grey curves) and those from CST (black curves) for x = -5 mm are plotted Fig. 16.

Fig. 16. Comparisons between Oy-profiles of the H-field components computed with CSTsoftware and those obtained from the proposed method

The imperfection of the results presented here are due to the numerical errors mainly caused

by the solver and the meshing inaccuracies. Note that one evaluates relative errors of about10 % for |Hx|, |Hy| and |Hz|. Despite the apparent difference between the results from the

method under investigation and the commercial EM-tool CST-computations, once again,very good correlations between the profiles of the H-field components are realized by

considering the data recorded in the vertical cut-plane equated by x = -5 mm as explained in

Fig. 16.

In nutshell, the computation results exposed in this paper reveal the effectiveness and theoperability of the method developed for the case of elementary magnetic dipoles and also by

considering the NF EM-radiation of realistic use case electronic devices.

5. Concluding remarks

A computation method of transient NF EM-field radiated by electronic devices excited by a

complex wave or ultra-short duration transient signal is stated in this chapter. In addition to

the evanescent wave integration, the originality of the NF calculation method developed lies

on the consideration of the radiation deeming the UWB structures which is literally from DCto microwave frequency ranges. It is based on the convolution of the frequency-dependent


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EM-wave data with a transient excitation pulse current. A methodological analysis wasmade by taking into account a complex waveform of the transient pulse signal exciting theradiation source structure considered. It was explained how the frequency-bandwidth of thefrequency-dependent baseband EM-field must be chosen according to the excitation current

considered.

In order to demonstrate the relevance of the method investigated, it was first, implementedinto Matlab program and then, applied to the determination of the H-field radiated by amicrowave circuit in UWB. As consequence, the feasibility of the method was verified withtwo types of structures. First, with the semi-analytical calculation implemented in Matlab byconsidering the frequency- and time-dependent expressions of the magnetic NF radiated bya set of magnetic dipoles, an excellent agreement with the results from the calculationmethod developed were found. Then, further more practical analysis was performed withthe determination of transient H-NF from the frequency-dependent data computed with astandard commercial 3-D EM-tool. For this second test, the H-NF emitted by a low-pass

planar microstrip filter was treated. For both cases, the excitation current injected to thestructures was assumed as an ultra-short transient pulse having half-bandwidth lower than5 ns which presents a baseband frequency spectrum with bandwidth of about 2.5 GHz fromDC. With the examples of complex structures tested, very good agreement between thetransient H-field component maps and profiles was realized from the method proposed andthose directly calculated from the well-known standard tools and from classicalmathematical EM-formulae.

It is interesting to point up that the NF computation method introduced in this chapter isadvantageous in terms of:

1. Simplicity of the EM-field maps determination for any waveform of transient excitationeven with ultra-short duration which is very hard to simulate with most of commercialsimulation tools. The method developed can be used for the determination of the NFmaps in time-domain which is practically very difficult to measure in the realisticcontexts.

2. It is flexible for various types of excitation signals which can be expressed analyticallyand also from the realistic use case of disturbing signal generally met in EMC area(Wiles 2003, Liu, K. 2011, Hubing 2011).

3. It can be adapted also to different forms of electrical and electronic structures for low-and high-frequency applications. Globally speaking, it offers a possibility to work inUWB from DC to unlimited upper frequency limit.

4. One can achieve significant EM-field measurement in very short time-duration withbase band measured data in wide bandwidth.

However, its main drawback is the limitation in term of time step which depends on thefrequency range of the initial frequency-data considered and also the necessity of powerfulcomputer for the achievement of high accurate results.

In the next step of this work, we plane to extend this method to transpose in time-domainthe modelling of EM-radiation with the optimized association of elementary dipoles (Vives-Gilabert et al. 2009, Fernndez-Lpez et al. 2009). Then, we are hopeful that the methoddeveloped in this chapter is very helpful for EMC/EMI investigations of modernelectrical/electronic systems as the case of hybrid vehicle embedded circuits (Vye 2011,


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Trscher 2011, Kopp 2011) where the transient NF effects are susceptible to disturb thesystem functioning.

6. Appendix

This appendix contains two parts of theoretical parts concerning the transient NF radiatedby the elementary magnetic dipoles (Baum 1971 & 1976, Singaraju & Baum 1976, Ravelo etal. 2011a & 2011b, Lui Y. et al. 2011a & 2011b) and the bi-exponential signal processing.

6.1 Appendix A: Analytical study of the magnetic dipole radiation in time-domain

By definition, the magnetic dipole moment of the elementary circular loop shown earlier inFig. 3 (see section 3) is written as:

( , ) ( ) ( ) zMp r t p t r u= G G

, (A-1)

with ris the distance between the dipole centre and the pointM(r,,) as shown in Fig. 3. Byanalogy with the definition of the time-variant vector established by Hertz in 1892 (Hertz1892), the H-field components in the spherical coordinate system:

( , , , ) ( , , , ) ( , , , )rrH H r t u H r t u H r t u = + +JJG G G G

, (A-2)

are expressed as:

2

( ) ( )cos( ) 1( , , , )

2r

p pH r t

r v tr

= +

, (A-3)

2

2 2 2

( ) ( ) ( )sin( ) 1( , , , )

4

p p prH r t

r v tr v t

= + +

, (A-4)

( , , , ) 0H r t = , (A-5)

where v is the wave-velocity, and is the time delayed variable which is defined as/t r v = . One underlines that the magnetic dipole is also an Hertzian dipole so that

( , ) / 0i r t r = . In the frequency domain, the spherical coordinate of the H-field componentformulas radiated by the magnetic dipole pictured in Fig. 3 which is supposed flowed by anharmonic current with amplitude IMdenoted:

2( ) j ft j tM MI f I e I e = = , (A-6)

are written as (Balanis 2005):

( )2

3( , , , ) 1 cos( )

2

jkrMr

I aH r f jkr e

r

= + , (A-8)

22 2

3

sin( )( , , , ) (1 )

4

jkrMI aH r f jkr k r er

= + , (A-9)


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( , , , ) 0H r f = , (A-10)

wherej is the complex number 1 and the real 2 /k f v= expresses the wave number atthe considered frequency f. Then, through the classical relationship between the spherical

and cartesian coordinate systems, one can determine easily the expressions of thecomponents Hx, Hy and Hz.

6.2 Appendix B: Spectrum analysis of bi-exponential signal

A bi-exponential form signal with parameters 1 and 2 is analytically expressed as:

1 2/ /( ) ( )t tMi t I e e = . (B-1)

The analytical Fourier transform expression of this current is written as:

1 2

1 2

( ) ( )1 1

MI Ij j

=

+ +, (B-2)

with is the angular frequency. This yields the signal frequency spectrum formulation

expressed as:

1 2

2 2 2 21 2

( )(1 )(1 )

MI I

=

+ +. (B-3)

To achieve at least 95-% of excitation signal spectrum energy, the frequency-data should berecorded in baseband frequency range with angular frequency bandwidth equal to:

2 2 2 2 2 2 21 2 1 2 1 2

95%1 2

( ) 1600

2

+ = . (B-4)

7. Acknowledgment

Acknowledgement is made to EU (European Union) and Upper Normandy region for the

support of these researches. These works have been implemented within the frame of the

Time Domain Electromagnetic Characterisation and Simulation for EMC (TECS) project

No 4081 which is part-funded by the Upper Normandy Region and the ERDF via the

Franco-British Interreg IVA programme.

8. References

Adada, M. (2007). High-Frequency Simulation Technologies-Focused on Specific High-Frequency Design Applications. Microwave Engineering Europe, (Jun. 2007), pp. 16-

17

Agilent EEsof EDA. (2008). Overview: Electromagnetic Design System (EMDS), (Sep. 2008)

[Online]. Available from:

http://www.agilent.com/find/eesof-emds


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Ansoft corporation. (2006). Simulation Software: High-performance Signal and Power Integrity,Internal Report

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